Addendum to “A formulation of volumetric growth as a mechanical problem subjected to non-holonomic and rheonomic constraint”

Grillo, Alfio; Stefano, Salvatore Di

doi:10.1177/10812865231178291

Cited by 7 publications

(1 citation statement)

References 7 publications

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“…The implementation of a variational formulation represents an elegant and effective approach to comprehending the relationship between the evolution of bone micro-architecture and the underlying physical laws that govern their behavior. This technique proves particularly advantageous in addressing complex problems, such as those encountered in studying bones and growth phenomena (Grillo and Di Stefano 2023 ; Grillo and Di Stefano 2023 ). This powerful approach can also be used to address challenging problems, such as those involving damage, that are intrinsically connected to the evolution of bone tissue (Placidi et al.…”

Section: Introductionmentioning

confidence: 99%

An orthotropic continuum model with substructure evolution for describing bone remodeling: an interpretation of the primary mechanism behind Wolff’s law

Giorgio,

dell’Isola,

Andreaus

et al. 2023

Biomech Model Mechanobiol

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We propose a variational approach that employs a generalized principle of virtual work to estimate both the mechanical response and the changes in living bone tissue during the remodeling process. This approach provides an explanation for the adaptive regulation of the bone substructure in the context of orthotropic material symmetry. We specifically focus upon the crucial gradual adjustment of bone tissue as a structural material that adapts its mechanical features, such as materials stiffnesses and microstructure, in response to the evolving loading conditions. We postulate that the evolution process relies on a feedback mechanism involving multiple stimulus signals. The mechanical and remodeling behavior of bone tissue is clearly a complex process that is difficult to describe within the framework of classical continuum theories. For this reason, a generalized continuum elastic theory is employed as a proper mathematical context for an adequate description of the examined phenomenon. To simplify the investigation, we considered a two-dimensional problem. Numerical simulations have been performed to illustrate bone evolution in a few significant cases: the bending of a rectangular cantilever plate and a three-point flexure test. The results are encouraging because they can replicate the optimization process observed in bone remodeling. The proposed model provides a likely distribution of stiffnesses and accurately represents the arrangement of trabeculae macroscopically described by the orthotropic symmetry directions, as supported by experimental evidence from the trajectorial theory.

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Section: Introductionmentioning

confidence: 99%

An orthotropic continuum model with substructure evolution for describing bone remodeling: an interpretation of the primary mechanism behind Wolff’s law

Giorgio,

dell’Isola,

Andreaus

et al. 2023

Biomech Model Mechanobiol

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Effective elasto‐(visco)plastic coefficients of a bi‐phasic composite material with scale‐dependent size effects

Giammarini,

Ramírez‐Torres,

Grillo

2024

Math Methods in App Sciences

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We employ the theory of asymptotic homogenization (AH) to study the elasto‐plastic behavior of a composite medium comprising two solid phases, separated by a sharp interface and characterized by mechanical properties, such as elastic coefficients and “initial yield stresses” (i.e., a threshold stress above which remodeling is triggered), that may differ up to several orders of magnitude. We speak of “plastic” behavior because we have in mind a material behavior that, to a certain extent, resembles plasticity, although, for biological systems, it embraces a much wider class of inelastic phenomena. In particular, we are interested in studying the influence of gradient effects in the remodeling variable on the homogenized mechanical properties of the composite. The jump of the mechanical properties from one phase to the other makes the composite highly heterogeneous and calls for the determination of effective properties, that is, properties that are associated with a homogenized “version” of the original composite, and that are obtained through a suitable averaging procedure. The determination of the effective properties results convenient, in particular, when it comes to the multiscale description of inelastic processes, such as remodeling in soft or hard tissues, like bones. To accomplish this task with the aid of AH, we assume that the length scale over which the heterogeneities manifest themselves is several orders of magnitude smaller than the characteristic length scale of the composite as a whole. We identify both a fine‐scale problem and a coarse‐scale problem, each of which characterizes the elasto‐plastic dynamics of the composite at the corresponding scale, and we discuss how they are reciprocally coupled through a transfer of information from one scale to the other. In particular, we highlight how the coarse‐scale plastic distortions influence the fine‐scale problem. Moreover, in the limit of negligible hysteresis effects, we individuate two viscoplastic effective coefficients that encode the information of the two‐scale nature of the composite medium in the upscaled equations. Finally, to deal with a case study tractable semi‐analytically, we consider a multilayered composite material with an initial yield stress that is constant in each phase. Such investigation is meant to contribute to the constitution of a robust framework for devising the effective properties of hierarchical biological media.

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Reconciling Kozlov’s vakonomic method with the traditional non-holonomic method: solution of two benchmark problems

Pastore,

Giammarini,

Grillo

2024

Acta Mech

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Addendum to “A formulation of volumetric growth as a mechanical problem subjected to non-holonomic and rheonomic constraint”

Cited by 7 publications

References 7 publications

An orthotropic continuum model with substructure evolution for describing bone remodeling: an interpretation of the primary mechanism behind Wolff’s law

An orthotropic continuum model with substructure evolution for describing bone remodeling: an interpretation of the primary mechanism behind Wolff’s law

Effective elasto‐(visco)plastic coefficients of a bi‐phasic composite material with scale‐dependent size effects

Reconciling Kozlov’s vakonomic method with the traditional non-holonomic method: solution of two benchmark problems

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